How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# CMI Entrance 2019 - Answer Key, Sequential Hints

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This post contains questions from Chennai Mathematical Institute, CMI Entrance 2019, Sequential hints, answer key, solutions. Our team will upload the solutions. Problems are contributed by students.

## Problems

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1. Suppose $\frac{8^x + 27^x}{12^x + 18 ^x} = 7/6$. Find x.

2. There is a parallelogram ABCD. O is any point in the interior such that angle AOB + angle DOC = 180. Prove that angle ODC= angle OBC

3. Find the number of natural number n for which $n^6+n^4+1$ is a perfect square

4. $\phi_k (m) = \phi_1 (\phi_{k-1} (m))$  and $\phi_1 (m)$ is number of divisors of m . Find k for which $\phi_k (2019^2019) =2$

5. If there exists a calculator with 12 buttons, 10 being the buttons for the digits and A and B being two buttons being processes where if n is displayed on the calculator if A is pressed it increases the displayed number by 1 and if B is pressed it multiplies n by 2 hence 2n. Hence find the minimum number of moves to get 260 from 1

6. Find the sum 1+111+11111+1111111+.....1....111(2k+1) ones

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## Faculty Team (for this section)

Founder - Faculty at Cheenta. Pursuing Ph.D. at University of Wisconsin, Milwaukee. USA Research Interest: Geometric Group Theory

Faculty - Admin at Cheenta Pursuing B.Stat at Indian Statistical Institute, India

Faculty at Cheenta Pursuing B.Sc. Math at Chennai Mathematical Institute, India Research Interest - Algebraic Geometry

Associate Faculty at Cheenta

Pursuing B.Stat from Indian Statistical Institute, India

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